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An Efficient Multi Quantile Regression Network with Ad Hoc Prevention of Quantile Crossing

Jens Decke, Arne Jenß, Bernhard Sick, Christian Gruhl

TL;DR

The paper tackles quantile crossing and computational cost in neural quantile regression by introducing SCQRNN, which integrates differentiable sorting into a composite quantile framework to guarantee non-crossing quantiles. It demonstrates a forward-pass complexity of $O(K L^2 + L T + T\log T)$, versus $O(T K L^2)$ for MCQRNN, yielding substantial efficiency especially when $T$ scales with $L$. Through extensive experiments on nine synthetic datasets and the U-bend design problem, SCQRNN shows competitive RMSE and reliability while converging faster than MCQRNN (and faster than CQRNN in training due to sorting). This method offers a scalable, energy-conscious approach for reliable uncertainty quantification in high-stakes domains, with potential integration into active design optimization and other HPC-enabled applications.

Abstract

This article presents the Sorting Composite Quantile Regression Neural Network (SCQRNN), an advanced quantile regression model designed to prevent quantile crossing and enhance computational efficiency. Integrating ad hoc sorting in training, the SCQRNN ensures non-intersecting quantiles, boosting model reliability and interpretability. We demonstrate that the SCQRNN not only prevents quantile crossing and reduces computational complexity but also achieves faster convergence than traditional models. This advancement meets the requirements of high-performance computing for sustainable, accurate computation. In organic computing, the SCQRNN enhances self-aware systems with predictive uncertainties, enriching applications across finance, meteorology, climate science, and engineering.

An Efficient Multi Quantile Regression Network with Ad Hoc Prevention of Quantile Crossing

TL;DR

The paper tackles quantile crossing and computational cost in neural quantile regression by introducing SCQRNN, which integrates differentiable sorting into a composite quantile framework to guarantee non-crossing quantiles. It demonstrates a forward-pass complexity of , versus for MCQRNN, yielding substantial efficiency especially when scales with . Through extensive experiments on nine synthetic datasets and the U-bend design problem, SCQRNN shows competitive RMSE and reliability while converging faster than MCQRNN (and faster than CQRNN in training due to sorting). This method offers a scalable, energy-conscious approach for reliable uncertainty quantification in high-stakes domains, with potential integration into active design optimization and other HPC-enabled applications.

Abstract

This article presents the Sorting Composite Quantile Regression Neural Network (SCQRNN), an advanced quantile regression model designed to prevent quantile crossing and enhance computational efficiency. Integrating ad hoc sorting in training, the SCQRNN ensures non-intersecting quantiles, boosting model reliability and interpretability. We demonstrate that the SCQRNN not only prevents quantile crossing and reduces computational complexity but also achieves faster convergence than traditional models. This advancement meets the requirements of high-performance computing for sustainable, accurate computation. In organic computing, the SCQRNN enhances self-aware systems with predictive uncertainties, enriching applications across finance, meteorology, climate science, and engineering.
Paper Structure (16 sections, 1 theorem, 17 equations, 5 figures, 1 table)

This paper contains 16 sections, 1 theorem, 17 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Related Work
  3. Quantile Regression Neural Network
  4. Composite Quantile Regression Neural Network
  5. Monotone Composite Quantile Regression Neural Network
  6. Differentiable Sorting
  7. Methodology
  8. Sorted Composite Quantile Regression Neural Network
  9. Theoretical Complexity Analysis
  10. Datasets
  11. Experiment 1
  12. Experiment 2
  13. Results and Discussion
  14. Experiment 1
  15. Experiment 2
  16. ...and 1 more sections

Key Result

proposition thmcounterproposition

Let $\hat{y}^{\mathbf{\tau}}=(\hat{y}^{\tau_1},...,\hat{y}^{\tau_T})$ be an estimate of the conditional quantile function at a point $x$ for $\mathbf{\tau}=(\tau_1,...,\tau_T)$. Let $\check{y}^{\mathbf{\tau}}=\mathcal{S}(\hat{y}^{\mathbf{\tau}})$ with $\mathcal{S}$ being the sorting operator. Then t Moreover, if sorting is nontrivial: $\check{y}^{\mathbf{\tau}}\neq\hat{y}^{\mathbf{\tau}}$ the ineq

Figures (5)

  • Figure 1: Illustration of the MCQRNN, SCQRNN and CQRNN
  • Figure 2: Visualization of the datasets used in this article
  • Figure 3: Test RMSE for four models assessed across three example functions, each augmented with three distinct error functions. Examples 0 and 1 are mapped to the left axis, whereas example 2 is scaled to the right axis
  • Figure 4: Test Overall Reliability for four models assessed across three example functions, each modified with three distinct error functions.
  • Figure 5: Exemplarily chosen validation curves of a single simulation run.

Theorems & Definitions (1)

  • proposition thmcounterproposition